reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;

theorem Th4:
  (for n be Nat holds seq.n >= a) implies inf seq >= a
proof
  assume
A1: for n be Nat holds seq.n >= a;
  now
    let x be ExtReal;
    assume x in rng seq;
    then ex z be object st z in dom seq & x = seq.z by FUNCT_1:def 3;
    hence x >= a by A1;
  end;
  then a is LowerBound of rng seq by XXREAL_2:def 2;
  hence thesis by XXREAL_2:def 4;
end;
